# How did we get the general solution of heat equation for ends of the bar insulated case?

How did we get the general solution of $$u=c_1c_3 \cos{(\frac{n \pi x}{l})} e^{-\frac{n^2 \pi^2 c^2 t}{l^2}}$$ to be$$u(x,t)=\frac{1}{2}A_0+\sum_n A_n \cos{(\frac{nπx}{l})} e^{-n^2 \pi^2 c^2 t/l^2}$$

what do i already know? i know that the general solution we have written is half range cosine series. but, how can we express that u in terms of cosine series.. i tried putting n=0 and didn't get A0/2...though i got the another part by putting n from 1 to...n. Am I missing some concepts? Please remind me if I am.

• Is the first equation supposed to be a differential equation? I fixed the notation but in the original notation, there was no indication of anything related to differentials. Please rewrite the equations to correspond to the problem you have. – Matti P. Jul 30 '19 at 6:27
• it is a solution of a partial differential heat equation math.ubc.ca/~peirce/M257_316_2012_Lecture_11.pdf – nawab_shazad Jul 30 '19 at 6:34
• Which equations are you referring to, in the paper? I don't see the first equation ... – Matti P. Jul 30 '19 at 6:35